Estimate (REF ) is understood as a gain of integrability for the collision operator since the \(L^{2}\) -norm of \(Q^{+}(g,\cdot )\)  is estimated by the lower \(L^{\frac{2d}{2d-1}}\) -norm when considering a fixed integrable first entry \(g\) . One can use Riesz–Thorin interpolation theorem to extend this estimate to the \(L^{p}\) -spaces when combined with the convolution estimates of Theorem REF , see for instance [1]}, [2]}.
